Under the Standard Model all fundamental particles can be broken down into two groups, fermions that make up matter, and bosons that exchange the forces acting on matter. Due to the physics of the theory, almost all of the behaviour of the universe can be explained based on this handful of particles.
Fermions themselves further break down into two generations, that is, each fermion comes in a variety of three subtypes of increasing mass. For instance one of the most commonly known fermions is the electron, which also has two other less-well-known subtypes, the muon and tau. Fermions also come in two versions for each generation, with differing electric charge. A graph of all the fermions in the Standard Model is quite small. It contains the three generations of quarks and leptons, each broken down into two partners with differing charge.
On the other hand the bosons come in groupings that are nowhere near as "neat", including four distinct types, with subgroups containing anywhere from one to sixteen members. In addition there appears to be no generational structure, the photon only comes in one type for instance, and although it has partners in the W and Z particles, they don't really match up with anything in the fermion side.
The discrepancy between the "clean" fermion side and "messy" boson side has long been one of the most bothersome points of the Standard Model.
Supersymmetry suggests that that every fundamental fermion has a "superpartner" which is a boson, and similarly every fundamental boson has a superpartner which is a fermion. That is, there is a complete symmetry across the entire model, with three generations of bosons for instance.
It turns out that none of the particles in the Standard Model can be superpartners of each other, so if supersymmetry is correct there must be at least as many extra particles to discover as there are in the Standard Model. The simplest possibility consistent with the Standard Model is the Minimal Supersymmetric Standard Model[?] (MSSM).
A possibility in some supersymmetric models is the existence of very heavy stable particles called neutralinos[?] or photinos[?] which would interact very weakly with normal matter. These would be possible candidates for dark matter.